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7/29/2019 no from comples

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ESC101NFundamentals of Computing

Arnab Bhattacharya

[email protected]

Indian Institute of Technology, Kanpurhttp://www.iitk.ac.in/esc101/

1st

semester, 2010-11

Tue, Wed, Fri 0800-0900 at L7

Arnab Bhattacharya ([email protected]) ESC101N 2010 I 1 / 6
http://localhost/var/www/apps/conversion/tmp/scratch_5/[email protected]://www.iitk.ac.in/esc101/http://www.iitk.ac.in/esc101/http://localhost/var/www/apps/conversion/tmp/scratch_5/[email protected]://find/


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Growth of functions

Efficiency of a program (or function) is measured by its timecomplexity

Time complexity is measured for large inputs

Input size is in the limit

Provides a way to study asymptotic complexity of functions

Generally, algorithms that are better for large inputs are better in

most cases except very small sized inputs

http://find/


3/21

Asymptotic upper bound: O-notation

O(g(n)) = {f(n) : c, n0 > 0 such that n n0, 0 f(n) cg(n)}

g(n) is an asymptotic upper bound of f(n)

n0n

Time f(n)

cg(n)

f(n) = O(g(n))

http://find/


4/21




f(n) = 1/2n2 3n = O(n2)

n0n

Time f(n)

cg(n)

f(n) = O(g(n))

http://find/


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f(n) = 1/2n2 3n = O(n2)

f(n) = 3n = O(n2)

n0n

Time f(n)

cg(n)

f(n) = O(g(n))

http://find/http://goback/


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f(n) = 1/2n2 3n = O(n2)

f(n) = 3n = O(n2)

f(n) = 6n3 = O(n2)

n0n

Time f(n)

cg(n)

f(n) = O(g(n))



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f(n) = 1/2n2 3n = O(n2)

f(n) = 3n = O(n2)

f(n) = 6n3 = O(n2)

o(g(n)): for any positive constant c> 0, 0 f(n) < cg(n)

f(n) is asymptotically smaller than g(n) if f(n) = o(g(n))

n0n

Time f(n)

cg(n)

f(n) = O(g(n))

http://find/


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Asymptotic lower bound: -notation

(g(n)) = {f(n) : c, n0 > 0 such that n n0, 0 cg(n) f(n)}

g(n) is an asymptotic lower bound of f(n)

n0n

Time

f(n)

cg(n)

f(n) = (g(n))

http://find/


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f(n) = 1/2n2 3n = (n2)

n0n

Time

f(n)

cg(n)

f(n) = (g(n))

http://find/


10/21




f(n) = 1/2n2 3n = (n2)

f(n) = 6n3 = (n2)

n0n

Time

f(n)

cg(n)

f(n) = (g(n))

http://find/


11/21




f(n) = 1/2n2 3n = (n2)

f(n) = 6n3 = (n2)

f(n) = 3n = (n2)

n0n

Time

f(n)

cg(n)

f(n) = (g(n))

http://find/


12/21




f(n) = 1/2n2 3n = (n2)

f(n) = 6n3 = (n2)

f(n) = 3n = (n2)

(g(n)): for any positive constant c> 0, 0 cg(n) < f(n)

f(n) is asymptotically larger than g(n) if f(n) = (g(n))

n0n

Time

f(n)

cg(n)

f(n) = (g(n))



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Asymptotic tight bound: -notation

(g(n)) = {f(n) : c1, c2, n0 > 0 such thatn n0, 0 c1g(n) f(n) c2g(n)}

g(n) asymptotically bounds f(n) from both above and below

c2g(n)

c1g(n)

n0

f(n)

n

Time

f(n) = (g(n))

http://find/


14/21



g(n) asymptotically bounds f(n) from both above and belowf(n) = 1/2n2 3n = (n2)

Choose c1 = 1/14, c2 = 1/2, n0 = 7

Other choices may also exist

c2g(n)

c1g(n)

n0

f(n)

n

Time

f(n) = (g(n))

http://find/


15/21



g(n) asymptotically bounds f(n) from both above and belowf(n) = 1/2n2 3n = (n2)

Choose c1 = 1/14, c2 = 1/2, n0 = 7

Other choices may also exist

f(n) = 6n3 = (n2)

Requires n c2/6 which is not true for all large n since c2 is constant

c2g(n)

c1g(n)

n0

f(n)

n

Time

f(n) = (g(n))

http://find/


16/21

Examples

Searching an array of size nfor ( i = 0 ; i < n ; i + + )

if ( a[ i] == key )return i ;


E l
http://find/


17/21

Examples



Average time is n/2 = O(n)

Also (n) and (n)


E l
http://find/


18/21

Examples




Also (n) and (n)

Transposing a matrix of size n n

for ( i = 0 ; i < n ; i + + )for ( j = 0; j < n ; j ++)

b [j ][ i ] = a [ i][ j ];


E l
http://find/


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Examples




Also (n) and (n)


for ( i = 0 ; i < n ; i + + )for ( j = 0; j < n ; j ++)

b [j ][ i ] = a [ i][ j ];

O(n2)

Also (n2) and (n2)


E l
http://find/


20/21

Examples


if ( a[ i] == key )

return i ;


Also (n) and (n)


for ( i = 0 ; i < n ; i + + )for ( j = 0; j < n ; j ++)

b [j ][ i ] = a [ i][ j ];

O(n2)

Also (n2) and (n2)

Accessing an element of an array of size na [i ] = valu e ;


E l
http://find/


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Examples


if ( a[ i] == key )

return i ;


Also (n) and (n)


for ( i = 0 ; i < n ; i + + )for ( j = 0; j < n ; j ++)

b [j ][ i ] = a [ i][ j ];

O(n2)

Also (n2) and (n2)

Accessing an element of an array of size na [i ] = valu e ;

Constant time, denoted as O(1) since it only requires addresscalculation and does not depend on the size of the array

Also (1) and (1)

http://find/

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